Complex Variables and Applications

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Author : James Ward Brown, Ruel V. Churchill

Publication : McGraw Hill Education 

Publishing Year : April 16, 2014

Edition : 8

Pages : 860

ISBN : 9789339205157


Complex Variables and Applications, 8e will serve, just as the earlier editions did, as a textbook for an introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier editions. The text is designed to develop the theory that is prominent in applications of the subject. You will find a special emphasis given to the application of residues and conformal mappings. To accommodate the different calculus backgrounds of students, footnotes are given with references to other texts that contain proofs and discussions of the more delicate results in advanced calculus. Improvements in the text include extended explanations of theorems, greater detail in arguments, and the separation of topics into their own sections.


1 Complex Numbers

Sums and Products

Basic Algebraic Properties

Further Properties

Vectors and Moduli

Complex Conjugates

Exponential Form

Products and Powers in Exponential Form

Arguments of Products and Quotients

Roots of Complex Numbers


Regions in the Complex Plane

2 Analytic Functions

Functions of a Complex Variable


Mappings by the Exponential Function


Theorems on Limits

Limits Involving the Point at Infinity



Differentiation Formulas

Cauchy–Riemann Equations

Sufficient Conditions for Differentiability

Polar Coordinates

Analytic Functions


Harmonic Functions

Uniquely Determined Analytic Functions

Reflection Principle

3 Elementary Functions

The Exponential Function

The Logarithmic Function

Branches and Derivatives of Logarithms

Some Identities Involving Logarithms

Complex Exponents

Trigonometric Functions

Hyperbolic Functions

Inverse Trigonometric and Hyperbolic Functions

4 Integrals

Derivatives of Functions w(t)

Definite Integrals of Functions w(t)


Contour Integrals

Some Examples

Examples with Branch Cuts

Upper Bounds for Moduli of Contour Integrals


Proof of the Theorem

Cauchy–Goursat Theorem

Proof of the Theorem

Simply Connected Domains

Multiply Connected Domains

Cauchy Integral Formula

An Extension of the Cauchy Integral Formula

Some Consequences of the Extension

Liouville’s Theorem and the Fundamental Theorem of Algebra

Maximum Modulus Principle

5 Series

Convergence of Sequences

Convergence of Series

Taylor Series

Proof of Taylor's Theorem


Laurent Series

Proof of Laurent's Theorem


Absolute and Uniform Convergence of Power Series

Continuity of Sums of Power Series

Integration and Differentiation of Power Series

Uniqueness of Series Representations

Multiplication and Division of Power Series

6 Residues and Poles

Isolated Singular Points


Cauchy’s Residue Theorem

Residue at Infinity

The Three Types of Isolated Singular Points

Residues at Poles


Zeros of Analytic Functions

Zeros and Poles

Behavior of Functions Near Isolated Singular Points

7 Applications of Residues

Evaluation of Improper Integrals


Improper Integrals from Fourier Analysis

Jordan’s Lemma

Indented Paths

An Indentation Around a Branch Point

Integration Along a Branch Cut

Definite Integrals Involving Sines and Cosines

Argument Principle

Rouché’s Theorem

Inverse Laplace Transforms


8 Mapping by Elementary Functions

Linear Transformations

The Transformation w = 1/z

Mappings by 1/z

Linear Fractional Transformations

An Implicit Form

Mappings of the Upper Half Plane

The Transformation w = sin z

Mappings by z2 and Branches of z1/2

Square Roots of Polynomials

Riemann Surfaces

Surfaces for Related Functions

9 Conformal Mapping

Preservation of Angles

Scale Factors

Local Inverses

Harmonic Conjugates

Transformations of Harmonic Functions

Transformations of Boundary Conditions

10 Applications of Conformal Mapping

Steady Temperatures

Steady Temperatures in a Half Plane

A Related Problem

Temperatures in a Quadrant

Electrostatic Potential

Potential in a Cylindrical Space

Two-Dimensional Fluid Flow

The Stream Function

Flows Around a Corner and Around a Cylinder

11 The Schwarz–Christoffel Transformation

Mapping the Real Axis onto a Polygon

Schwarz–Christoffel Transformation

Triangles and Rectangles

Degenerate Polygons

Fluid Flow in a Channel Through a Slit

Flow in a Channel with an Offset

Electrostatic Potential about an Edge of a Conducting Plate

12 Integral Formulas of the Poisson Type

Poisson Integral Formula

Dirichlet Problem for a Disk

Related Boundary Value Problems

Schwarz Integral Formula

Dirichlet Problem for a Half Plane

Neumann Problems



Table of Transformations of Regions



Data sheet


Specific References


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Complex Variables and Applications